I would like to suggest the three volume set by Dubrovin, Fomenko, Novikov (Modern Geometry--Methods and Applications) as a supplementary reference. They have a somewhat unique style and approach to the subject. The first volume begins with surfaces, the second volume goes on to manifolds. They give examples from physics along the way, which some may find interesting/useful.

I also like the chatty, informal style of M. Berger. He doesn't shy away from giving informal descriptions of ideas and motivations behind definitions. Perhaps most books try to do this, but Berger is particularly generous with it, and good at it, in my opinion. I have his book A Panoramic View of Riemannian Geometry in mind--this may not be the best place to learn about differential geometry for the first time, but I think some of his insights/comments would be useful even for beginners (see, e.g., pp 143-151). I think he has other, more elementary books on geometry but I don't have the references right now.